J Optim Theory Appl (2008) 137: 641–673
DOI 10.1007/s10957-007-9339-1

Monotone Comparative Statics: Geometric Approach
B.H. Strulovici · T.A. Weber

Published online: 29 December 2007
© Springer Science+Business Media, LLC 2007

Abstract We consider the comparative statics of solutions to parameterized optimization problems. A geometric method is developed for finding a vector field that,at each point in the parameter space, indicates a direction in which monotone comparative statics obtains. Given such a vector field, we provide sufficient conditions
under which the problem can be reparameterized on the parameter space (or a subset
thereof) in a way that guarantees monotone comparative statics. A key feature of our
method is that it does not require the feasible set to be a lattice and works in the absence of the standard quasi-supermodularity and single-crossing assumptions on theobjective function. We illustrate our approach with a variety of applications.
Keywords Change of parameters · Parameterized optimization problems ·Single-crossing · Supermodularity
1 Introduction
In many model-based optimization problems, important insights can be derived bycomparing solutions for different parameter values. The model’s parameters are exCommunicated by D.G. Luenberger.
We are grateful to Kenneth Arrow, Darrell Duffie, David Luenberger, Paul Milgrom, John Quah, andPete Veinott for helpful comments.
Research in part supported by a David Morgenthaler II Faculty Scholar Award.
B.H. Strulovici
Nuffield College, Oxford University, Oxford, UKe-mail: bruno.strulovici@economics.ox.ac.uk
T.A. Weber ()Department of Management Science and Engineering, 442 Terman Engineering Center,
Stanford University, Stanford, CA 94305-4026, USAe-mail: webert@stanford.edu

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ogenously specified and often can be varied for analysis purposes, while its variables,
which constitute the building blocks for its predictions, are endogenously determinedby (i) imposed model relations and (ii) parameter values. For instance, an economic
model might be concerned with a firm’s optimal production of widgets (e.g., in termsof capital and labor requirements), given both a production function relating output
to inputs and a set of prices (e.g., the market price for widgets, the cost of capital,
and an average wage rate). In the neoclassical tradition the model would then impose
maximization of the firm’s profit to determine optimal amounts of factor inputs asa function of their respective prices. More generally, if a model’s predictions can be
expressed as an optimal action in some finite-dimensional space, then comparative
statics studies the direction in which the optimal action changes as a result of some
disturbance in the values of the model’s parameters [1]. Thus, in our example, normalizing the price of the firm’s output to one, the optimal choice of inputs criticallydepends on the prices of the production factors which are this model’s parameters.
Sensitivity and perturbation analysis has been the traditional mathematical approach to analyze changes of objective values and solutions with respect to parameters [2, 3]. The associated methods focus on establishing regularity properties
and generally do not consider the monotonicity properties of solutions, which areof prime importance for decision makers in many applied optimization problems.
In this paper we consider the possibility of guaranteeing monotone behavior of solutions through an appropriate reparameterization of the problem. The key questionof monotone comparative statics is to determine under what conditions the model
predictions vary monotonically with the parameters [4, 5]. General answers to thisquestion for optimal actions chosen from feasible sets, which are usually assumed tobe lattices satisfying a set-monotonicity requirement with respect to the parameters,
are provided by [4–6]. They provide a necessary and sufficient condition for optimal
actions to exhibit monotone comparative statics with respect to the parameters. For
the special case of our neoclassical production decision problem, where the production function is independent of the factor prices, monotone comparative statics (i.e.,
inputs nonincreasing in prices) obtain if and only if the production function is supermodular, which—assuming twice continuous differentiability—amounts to requiringthat all cross-partial derivatives of the production function are nonnegative.
The practical importance of monotone comparative statics that justifies its widespread use, particularly in economics, lies in the fact that robust insights can be obtained in the absence of an analytical solution to the model: the monotonicity of
optimal actions in parameters is guaranteed if the system’s objective function satisfies certain easy-to-check requirements. In addition, monotonicity of optimal actions
can yield useful rules of thumb for decision makers and thus help in arriving at “optimally imperfect decisions” [7]. Clearly, in our production example (which is examined more closely in Sect. 5.2) it would be helpful for the firm to be able to translate
price movements immediately (without any further computations) into appropriate
input changes which at the very least vary in the right direction, even when its production function is not supermodular due to anticomplementarities between factor
inputs. Unfortunately, the currently available theory on monotone comparative statics returns negative results in situations where the aforementioned characterizationof monotone comparative statics by [6] fails. We argue that this failure is often due

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to the fact that the parameterization of the problem is taken as given.1 Indeed, ourresults indicate that it may be possible to achieve monotonicity of solutions in newparameters that are obtained by a one-to-one mapping from the original parameter
space. In fact, for problems with a smooth analytical structure we show that, provided with sufficiently precise knowledge about the location of an optimal action in
the action space, it is possible to find a reparameterization that achieves monotone
comparative statics in any single component of the decision.
Our central goal is to provide a new method for achieving monotone comparative statics of solutions to parameterized optimization problems, first by relaxing andthereby generalizing the standard monotone comparative statics problem, and second,by providing a systematic way to reparameterize the problem such that monotone
comparative statics can be achieved.2 In addition to addressing cases where the standard results do not apply, our method can, by offering a new description of the parameter space, shed light on important relations between decision variables and parameters of economic problems.
The approach developed here builds on tools in differential geometry, and we thus
require models with a smooth structure (e.g., a parameterized optimization problem
with a twice continuously differentiable objective function), even though—as we are
well aware—none of our statements fundamentally depends on the differentiable
structure. All of our results can also be expected to hold if the problem is suitably
discretized; nevertheless, we prefer to adopt a differentiable approach for ease of
exposition, since then the tools of differential geometry can be applied seamlessly.
We decompose the problem of changing problem parameters to achieve monotone
comparative statics (MCS) into two parts. First, the decision maker needs to solve a
local MCS problem by finding for each point t in the parameter space T a direction
vector v(t) (i.e., an element of the tangent space of T at t) that would increase theoptimal action x(t) if parameters were to be locally changed from t in the directionof v(t). If the location of x(t) is not known precisely, as is generally the case, thenthe direction v(t) must be such that it induces local monotonicity with respect to all
points in a subset R(t) of the action space X which is known to contain the optimal
action x(t), given the possibility that any point in R(t) might turn out to be optimal.
A full solution to this problem consists in a vector field v(t), defined for all valuesof t in the parameter space. Second, given the vector field v(t), the decision makerneeds to find a reparameterization solving the global MCS problem. We show that
this can always be achieved locally through “rectification” of the vector field. Undera few additional conditions rectification can also be achieved globally, leading to thedesired global MCS reparameterization of the decision problem.
The paper proceeds as follows. In Sect. 2 we introduce the problem of obtaining
monotone comparative statics (i.e., monotone dependence of solutions on parameters) for parameterized optimization problems, both from a local and a global view1 This is true in the literature, except for some rare cases where trivial reparameterizations such as a changeof sign or other simple ad hoc reparameterizations are chosen under very special circumstances, e.g., by [8]in a network flow problem.
2 Even though not explicitly developed, our methods apply equally to equilibrium problems, by replacing

the first-order necessary optimality conditions of the optimization problem with the equations specifying
equilibria.

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point. When considered locally, obtaining monotone comparative statics corresponds
to finding directions in the parameter space in which solutions to the optimization
problem are nondecreasing in parameters. A solution to the local monotone comparative statics problem is provided in Sect. 3 using a vector field method. Subsequently, in Sect. 4, we take a more global perspective: having obtained a vector fieldof monotone comparative statics directions defined at each point of the parameter
space, we demonstrate that it is possible (at least locally) to change the parametersof the optimization problem, i.e., to reparameterize it such that monotone comparative statics of the solutions of the reparameterized problem obtain. We show that thereparameterization can be global if a hyperplane can be found that is transverse to avector field that solves the local MCS problem at each point of the parameter space.
To illustrate our results we discuss a number of economic applications in Sect. 5 andconclude with a discussion and directions for further research in Sect. 6.

2 Problem Formulation
We consider a decision maker who, given a parameter value t ∈ T , aims to selectan element x(t) of an action space X so as to maximize her objective function f :X × T → R. She thus tries to solve the parameterized optimization problem
x(t) ∈ arg max f (x, t),
x∈X

(1)

where X is a n-dimensional compact manifold3 and T is an m-dimensional compact manifold. Provided that a solution to (1) exists,4 the decision maker is interested
in the comparative statics of the maximizer x(t) as t ∈ T varies. More generally,
we assume that the decision maker is concerned with the behavior of the composition ϕ ◦ x(t) = ϕ(x(t)), where ϕ : X → Rd (with 1 ≤ d ≤ n) is an evaluation functionthat the decision maker uses to assess any solution x(t) of (1). For example, if the decision maker is interested only in the comparative statics of the first component of
the maximizer x(t) = (x1 , . . . , xn )(t), she can choose ϕ(x) = x1 . From Milgrom andShannon’s [6] monotonicity theorem, we know that if ϕ(x) = x and X is a lattice,then ϕ ◦ x(t) = x(t) is increasing in t (on any given sublattice of X) if and only
if f is quasi-supermodular5 in x and satisfies the single-crossing property6 in (x, t).
3 If X lies in a lower-dimensional submanifold of Rn , the analysis can still be applied, but differential

calculus should be understood on this submanifold, and openness should be understood relative to thesubmanifold, cf. Sect. 3.5.
4 If for any parameter t ∈ T the function f (·, t) is continuous and X is bounded, a solution to the parame-terized optimization problem (1) exists in the closure of X by the Weierstrass theorem ([9], p. 540).

5 A real-valued function f defined on a lattice X ⊂ Rn is quasi-supermodular if f (x) ≥ (>)f (x ∧ y)

implies f (x ∨ y) ≥ (>)f (y), for all x, y in X , where x ∨ y = (max{x1 , y1 }, . . . , max{xn , yn }) and x ∧ y =(min{x1 , y1 }, . . . , min{xn , yn }). As its name suggests, quasi-supermodularity is a weaker condition thansupermodularity.

6 A real-valued function f defined on the product X × T of two partially ordered spaces has the single-

crossing property if, whenever x > x and t > t , f (x , t) ≥ (>)f (x, t) implies f (x , t ) ≥ (>)f (x, t ).
The single-crossing property is a weaker condition than supermodularity in (x, t).

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Conversely, for any objective function f that does not satisfy these conditions, x(t)cannot be nondecreasing on T . Nevertheless, despite this negative result using thestandard theory, it may be possible for the decision maker to at least find a path γt in
the parameter space, so that starting at a given t ∈ T the function ϕ ◦ x is nondecreasing along γt . Monotone comparative statics may thus be obtained following certain
directions in the parameter space. It is useful given any t ∈ T to state the decision
maker’s MCS problem (at t) in precisely these terms.
MCS Problem (at t) Given a continuously differentiable evaluation function ϕ anda parameter value t ∈ T , find a nonempty open interval It ⊂ R with 0 ∈ It and apath γt : It → T , such that ϕ ◦x(γt (λ)) is nondecreasing7 for all λ ∈ It and γt (0) = t.
If a solution (It , γt ) of the MCS problem at t is such that the path γt cannot
be extended in T , then we call the solution maximal. We can restrict our attention, without any loss of generality, to a maximal solution of the MCS problem at t.
Monotone comparative statics relative to an evaluation function ϕ and a parameterstarting value t obtain whenever the vector ϕ ◦ x is componentwise nondecreasingalong an appropriate path γt in the parameter space T . Along any such path, x(γt (λ))
solves (1) for all λ ∈ It . A solution to the MCS problem for all t ∈ T results in a
global flow θ (λ, t) = γt (λ), for which θ (0, t) = t andλ≤μ⇒ϕ ◦ x(θ (λ, t)) ≤ ϕ ◦ x(θ (μ, t)),(2)

for any λ, μ ∈ It . If it is possible to represent the global flow in the form
θ (λt , ψ(t)) = t,(3)

where λt is uniquely determined and ψ : T → P is a function that maps the parameter
space to a fixed (m − 1)-dimensional hypersurface P, transverse8 to the vector field
induced by the MCS paths γt , then using the new parameters s(t) = (s1 , . . . , sm )(t) =(λt , π(ψ(t))) guarantees monotone comparative statics of ϕ ◦ x(s) in s1 = λ, at leastm−1 (details are
locally, where π is a diffeomorphism
 from P onto a subset of R
provided in Sect. 4). Letting F = t∈T It × {t}, we formulate the global parameterchange problem accordingly.
Global MCS Reparameterization If the flow θ : F → T solves the MCS problem
everywhere in T , find a new parameterization s(t) = (s1 , . . . , sm )(t) = (λt , π(ψ(t)))
such that (3) is satisfied for all t ∈ T .
A global MCS reparameterization provides the decision maker with new problem parameters s = (s1 , . . . , sm ) that guarantee monotonicity of ϕ(x̂(s)) in the firstcomponent s1 ∈ I, wherex̂(s) = arg max fˆ(x, s)
x∈X

(4)

and fˆ(x, s) corresponds to the objective function f (x, t) after the parameter change.
7 A vector v(λ) is nondecreasing in λ if and only if each of its components is nondecreasing in λ.
8 See Assumption A6 for the precise definition.

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3 Solving the MCS Problem at t
Our goal is to find directions in which solutions of the parameterized optimization
problem (1) (or functions thereof) are increasing. For this we introduce a “pseudogradient” W (x, t) which mimics the gradient matrix ∇t x(t) = W (x(t), t) corresponding to all potential solutions x ∈ R(t) with respect to the parameters t on aset R(t) that is known to contain the actual solution x(t). We refer to R(t) as a“reduced feasible set,” for it is a subset of the set of all feasible actions X . The cardinality of R(t) is a measure of how much information the decision maker has about
the location of the solution to (1) at t. If for a given t ∈ T all row-vectors ∇t xi (t),i ∈ {1, . . . , n}, of the pseudo-gradient lie in the same m-dimensional half-space forall points of the reduced feasible set (a subset of X ), then a direction v(t) ∈ Rm \ {0}exists in which monotone comparative statics obtain locally. If such a direction v(t)can be found for all points t of the parameter space T , then the resulting vector
field v : T → Rm constitutes a solution to the MCS problem on T . The flow induced by this vector field can be used to obtain an MCS reparameterization of the
optimization problem (1), which is discussed in Sect. 4. In this section we first introduce a number of assumptions needed for the vector field method. We then provide
techniques to implement the method, and relate the vector field method to classic supermodularity results. Finally, we provide important methods to deal with problems
that contain equality and/or inequality constraints.
3.1 Assumptions
In order to use standard tools from differential geometry, we require that f be sufficiently smooth.
Assumption A1 (Smoothness) The objective function f is twice continuously differentiable in x and has continuous cross-derivatives with respect to each tuple (xi , tk ),for all 1 ≤ i ≤ n and all 1 ≤ k ≤ m.
Let us denote by
H (x, t) = ∇xx f (x, t) =n
 2
∂ f (x, t)∂xi ∂xj i,j =1

the Hessian matrix of f (·, t) evaluated at (x, t) and by
n,m
 2∂ f (x, t)K(x, t) = ∇xt f (x, t) =∂xi ∂tk i,k=1
the matrix of cross-derivatives of f between decision-variable and parameter components, evaluated at (x, t). In order to bypass (at least for now) any difficulties arisingfrom binding constraints at the optimal action, we will assume that the parameterized
optimization problem (1) possesses a unique unconstrained optimum. Constrained
optimization problems often can be restated equivalently so as to satisfy this assumption, and Sect. 3.5 is dedicated to this issue. We also emphasize that nothing in our
method requires that the feasible set X be a lattice (cf. also footnote 18).

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Assumption A2 (Existence and Uniqueness) For each t ∈ T , the parameterized optimization problem (1) has a unique solution x(t).
In general, the set of maximizers is guaranteed to be nonempty and in the interiorof X if, in addition to being continuous, f is coercive relative to X , in the sensethat for any t ∈ T there exists a point x̌(t) ∈ X such that9 f (x̌(t), t) ≥ sup f (∂X , t),(cf. [9], pp. 8, 540). If f (·, t) possesses multiple strict local extrema for some t ∈ T ,our results can be applied with respect to the comparative statics of each local maximum. Since X is open, Fermat’s Lemma ([10], Vol. I, p. 215) implies that each strict
local extremum c(t) (and in particular the unique global maximum x(t) guaranteedby Assumption A2) is a critical point of f (·, t), i.e., ∇x f (c(t), t) = 0. We denoteby C(t) the set of all critical points10 of f (·, t) in X at t,
C(t) = {x ∈ X : ∇x f (x, t) = 0}.
Sard [11, 12] has shown that the corresponding set of critical values f (C(t), t), whichis the image of C(t) under f , is of Lebesgue-measure zero.
Thus, if the decision maker can determine C(t), she might be able to find the
optimal action x(t) as the solution of a reduced optimization problem, x(t) =arg maxc∈C (t) f (c, t), using the first-order necessary optimality conditions. Assumption A2 also implies that for any t ∈ T at the unique global optimum x(t) the Hessian
matrix of f (·, t) is negative semidefinite, i.e., x(t) satisfies the second-order necessary optimality condition x(t) ∈ D(t), whereD(t) = {x ∈ X : H (x, t) ≤ 0}.
This allows the decision maker to further reduce the optimization problem combining
the first-order and second-order necessary optimality conditions and solvex(t) = arg max f (x, t),x∈R(t)(5)

where we refer to R(t) ⊆ X as a reduced feasible set; in this case R(t) = (C ∩ D)(t).
More generally, we refer to any subset R(t) of X which is guaranteed to contain thesolution x(t) of (1) as an admissible reduced feasible set. Any element of R(t) iscalled a reduced-feasible action. If the decision maker can determine an admissible
reduced feasible set R(t) (e.g., by using first- and second-order necessary optimality conditions), so that she is able to solve the (reduced) parameterized optimization
problem (5) on T , then the MCS problem always has a trivial solution, as will become clear below (cf. Theorem 3.1). Unfortunately, in many practical applications, aclosed-form solution of (1) is not possible, or the objective function is not perfectly
known by the decision maker (see [13] and Sect. 6). In that case, by constructing a
reduced feasible set R(t) ⊆ X that is guaranteed to contain the optimal action x(t)9 ∂X denotes the boundary of X .
10 Note that if f (·, t) has a critical point (i.e., X ∩ C(t) = ∅) and is strictly concave on X for all t ∈ T ,then Assumption A2 is automatically satisfied, since f (·, t) is necessarily single-peaked on X .

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J Optim Theory Appl (2008) 137: 641–673(e.g., by using heuristics related to the special structure of the problem), the decision
maker may still be able to solve the MCS problem without an explicit solution to the(equivalent) parameterized optimization problems (1) and (5). To obtain a solution to
the MCS problem when the optimal action x(t) can be only imperfectly localized in
the set R(t) ⊆ X , we require that all critical points of f (·, t) in R(t) be nondegenerate (i.e., such that the Hessian matrix of f (·, t) is nonsingular there).
Assumption A3 (Nondegeneracy) For any t ∈ T , the Hessian matrix H (x, t) is
nonsingular for all x ∈ R(t), for some reduced feasible set R(t) ⊆ X which contains x(t).
This assumption guarantees that the inverse H −1 (x, t) is well defined and continuous at any point (x, t) ∈ R(t) × T . Hence the expression −(H −1 K)(x, t), evaluatedat a point (x, t) possibly different from the optimal (x(t), t), is well-defined. Assumption A3 is automatically satisfied if the objective function is strictly concave.
Lemma 3.1 Under Assumptions A1–A3, the unique optimal solution x(t) of the parameterized optimization problem (1) is continuously differentiable on T . The corresponding Jacobi matrix is given by∂xi (t) n,m∇t x(t) == −(H −1 K)(x(t), t),(6)∂tk i,k=1
for all t ∈ T .
Proof By Assumption A2, a unique interior solution x(t) to the parameterized optimization problem (1) exists for all t ∈ T , satisfying H (x(t), t) ≤ 0 and∇x f (x(t), t) = 0.

(7)

By Assumption A1, we can differentiate (7) with respect to t (using the chain rule)and obtain∇t (∇x f (x(t), t)) = (∇xx f (x(t), t))(∇t x(t)) + ∇xt f (x(t), t) = 0,or equivalently
H (x(t), t)∇t x(t) + K(x(t), t) = 0,
for all t ∈ T . Since H (x(t), t) is nonsingular by Assumption A3, we get expression (6) after left-multiplication with H −1 (x(t), t) in the last equality. We now showthat ∇t x(t) is continuous. Since the maximizer x(t) is unique and the objective function f continuous, we have that, as a consequence of Berge’s ([14], p. 116) maximum
theorem, the maximizer x(t) is continuous in t (for it is upper-semicontinuous andsingle-valued). By virtue of Assumption A1 and the nonsingularity of H , all entriesof the matrix (H −1 K)(x(t), t) are well defined and, as a composition of continuousfunctions, also continuous. Hence, the Jacobi matrix ∇t x(t) on the left-hand side of(6) must also be continuous, which completes the proof.
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In order to study the monotonicity of ϕ ◦ x(t), we require some smoothness aswell as functional independence ([10], Vol. I, p. 508) of the evaluation function.
Assumption A4 (Functional Independence) The evaluation function ϕ is continuously differentiable on X and its Jacobi matrix∂ϕl (x) d,n(x) = ∇x ϕ(x) =∂xi l,i=1


has (full) rank d for any (reduced-)feasible action x.
Assumption A4 is not critical for our results and can (except for the smoothness
portion) be relaxed. Functional independence guarantees that the MCS problem is locally never trivial, since no two of ϕ’s components are collinear. If Assumptions A1–
A4 hold, then for any (reduced-)feasible tuple (x, t), we can define the (d ×m)-matrix
W (x, t) = − (x)H −1 (x, t)K(x, t),(8)which we term the pseudogradient of the MCS problem at (x, t). In analogy toLemma 3.1, it is easy to show that the pseudo-gradient evaluated at any optimizing decision-parameter tuple (x(t), t) describes the comparative statics of ϕ ◦ x(t)along paths parallel to the standard coordinates in the parameter space T , i.e.,∇t ϕ(x(t)) = W (x(t), t).

(9)

We say that the pseudo-gradient of the MCS problem is orientable at (x, t) if thecollection of all of its row vectors is a subset of a common half space of Rm . If for agiven t ∈ T the row vectors of W (x, t) lie in a common half space of Rm for all x ina set Y ⊆ X , then we say that the pseudogradient is orientable on Y at t.
Example 3.1 If m = n ≥ 1, φ(x) = x, and f (x, t) = x Qx − t x, where Q is a symmetric negative definite matrix, then = I , H = Q < 0, and K = −I . The corresponding pseudogradient is W = − H −1 K = Q−1 . This pseudogradient is orientable, since its n row vectors are linearly independent and span Rn . For example,
one can take the sum of the row vectors as a normal vector of the half-space that
contains all row vectors.
3.2 Vector Field Method
To find a path γt that solves the MCS problem at t ∈ T , our method requires eliciting some information about the direction of the gradient ∇t (ϕ(x(t))), given that thedecision maker knows only that x(t) lies in some reduced feasible set R(t) ⊆ X .
More specifically, we need to determine a direction forming acute angles with thepseudo-gradient of the MCS problem. Such a direction exists if the pseudo-gradientis orientable.

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Assumption A5 (Pseudogradient Orientability) There exists a continuously differentiable vector field v(t) ∈ Rm \ {0} such that, for each t ∈ T ,W (x, t)v(t) ≥ 0,

(10)

for all x ∈ R(t), where R(t) ⊆ X is an admissible reduced feasible set.
Under Assumption A5, v(t) defines a vector field on T and a phase diagram withpaths corresponding to the flow of this vector field.11 The key result of this section isthat ϕ ◦ x is nondecreasing along the paths.
Theorem 3.1 Under Assumptions A1–A5, let I ⊂ R be an open interval and letγ : I → T be a differentiable path such thatγ̇ (λ) = v(γ (λ)),
for all λ ∈ I. Then, ϕ(x(γ (λ))) is nondecreasing for all λ ∈ I.
Proof Since γ is differentiable on I, Lemma 3.1 implies (using Assumptions A1
and A2) that∇t x(γ (λ)) = −(H −1 K)(x(γ (λ)), γ (λ)).
Thus, using Assumption A4 and (9), the gradient of ϕ(x(γ (λ))) with respect to λ is
given by∇λ ϕ(x(γ (λ))) = W (x(γ (λ)), γ (λ))γ̇ (λ) = W (x(γ (λ)), γ (λ))v(γ (λ)).
By virtue of inequality (10) in Assumption A5, the latter expression is nonnegative. 
Under the assumptions of Theorem 3.1, we say that ϕ(x(t)) is nondecreasingalong the trajectories of v(t).
3.3 Implementation
Under Assumptions A1–A5 the vector field method can be implemented using thefollowing techniques. First, for any t ∈ T determine a reduced feasible set R(t) ⊆ X
that is guaranteed to contain the interior solution x(t) of the parameterized optimization problem (1). Second, to satisfy Assumption A5, verify that the pseudogradient W (x, t) is orientable on R(t). Note that for any given t ∈ T it may be possible
to achieve orientability by premultiplying the evaluation function ϕ by some diagonal (d × d)-matrix of the form M(t) = diag(m1 , . . . , md )(t) where ml (t) ∈ {−1, 1}
for all l ∈ {1, . . . , d}. Indeed if ϕ̂(x, t) = M(t)ϕ(x), then it is possible to find a matrix M(t) such that the pseudogradient Ŵ (x, t) corresponding to the modified evaluation function ϕ̂,Ŵ (x, t) = M(t)W (x, t) = −M(t) (x)H −1 (x, t)K(x, t),
11 The trajectories of a vector field v(t) exist and are unique on the whole domain T if the vector field is

Lipschitz there ([15], pp. 74–77), and in particular when v is continuously differentiable on T .

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Fig. 1 Solution to the MCS Problem at t for ϕ(x) = x1

is orientable at (x, t). In particular, if a matrix M can be found that is independent of t,
it may be advantageous for the analysis of the problem if the decision maker uses theevaluation function ϕ̂ instead of ϕ. Third, find a vector field v that satisfies (10). To
accomplish this, a systematic, algorithmic procedure to determine a vector field v(t)that is “maximally aligned” with the pseudogradient W (x, t) consists in solving the
maximin problem12
v(t) ∈ arg max
W
(x,
t),
v
.
(11)
min
min
l
m
v∈R :v=1 x∈R(t) 1≤l≤dWhen the assumptions of the min-max theorem [16] are satisfied, any solution v(t)to problem (11) also satisfies



max
min Wl (x, t), v .
v(t) ∈ arg min(12)mx∈R(t) v∈R :v=1 1≤l≤dMotivated by the minimax formulation (12), since W is orientable by assumption,
one obtains the set of candidate vector fields
Wd (x̂ d (t), t) W1 (x̂ 1 (t), t)
,
.
.
.
,
V=
,
(13)
Wd (x̂ d (t), t) t∈T
Wd (x̂ 1 (t), t)
where for any l ∈ {1, . . . , d} we have set
x̂ l (t) = arg min Wl (x, t).
x∈R(t)

One can now check whether some element of V is a suitable vector field on T or
possibly a subset thereof.
The above three steps can be iterated to tighten the reduced feasible set. It can also
be useful to consider only subsets of the parameter space T . Note that if an exact
12 W is the l-th row vector of W , and ·, · denotes the scalar product in the relevant Hilbert space.
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solution to the maximization problem (5) is known for some t ∈ T , then the MCS
problem at t has a solution if any only if W (x(t), t) is orientable at (x(t), t).
Example 3.2 Consider a firm that has the option to invest in a number x of geographically dispersed markets13 (e.g., cities in the US) at an increasing convex
cost C(x) ≥ 0. For simplicity, the market price p ∈ (0, 1) for the firm’s product isassumed to be the same in each market; it is announced nationally and is a parameterof the problem. Marginal production costs are zero. The demand in the each market
is D(p) = 1 − p, and the firm is risk averse with constant absolute risk aversion ρ. By
investing in x markets the firm also reaps an increasing concave side benefit B(x) ≥ 0(e.g., through real-estate transactions).14 With probability q ∈ (0, 1 − p) that is (atmost) proportional to the quantity sold in each market, the firm incurs a unit loss inany market due to a liability claim. The firm’s expected payoffs are approximately15
ρ(x, p, q) = xpD(p) + B(x) − E[L̃|q] − Var[L̃|q] − C(x)2ρq(1 − q)x + B(x) − C(x),= p(1 − p) − q −2where the random variable L̃ ∈ {0, 1, . . . , x} represents the firm’s total losses. Maximizing profits we thus obtain x ∗ (p, q) = ξ(p(1 − p) − q − ρq(1 − q)/2), where ξ isthe (increasing) inverse of C − B . Hence, the maximizer x(t) is increasing in the parameter t ∈ T = {(p, q) ∈ R2++ : p +q < 1} if and only if p(1−p)−q −ρq(1−q)/2
is increasing. The corresponding pseudogradient at the optimum is
W (x(t)) = [−1 + ρq − ρ/2, 1 − 2p] ξ (p(1 − p) − q − ρq(1 − q)/2),
so that, with v(t) = (−p, qρ/2), we obtain(4ρ − 1)ξ (t)ρξ (t) 2p+ p + q − 4pq ≥> 0,W (x(t)), v(t) =2
ρ
16
for all t ∈ T and ρ > 1/4. The vector field v solves the MCS problem on T . In
Example 4.1 we show how to obtain a global MCS reparameterization of the problem based on the vector field v, cf. Fig. 2. Let us remark that clearly in this example R(t) = {x(t)} if ξ −1 is known precisely (for any given B and C). However, ourconclusions can be obtained without further specifying the firm’s payoffs. Note that it
would have been possible to replace p(1 − p) − q − ρ(1 − q)q/2 by a scalar parameter λ, which, however, would decrease the resolution for a decision maker somewhat
13 We allow x to take non-integer values.
14 By imposing the Inada conditions C (0) < ∞ and B (0) = ∞ one can easily guarantee that the optimum
is interior, i.e., x(t) > 0, so that we can without loss of generality set X = (0, ∞).
15 The dependence ofon the parameter ρ is not explicitly noted. In fact, in this problem the comparative
statics with respect to ρ are obvious. We can thus use ρ itself in solving the MCS problem and finding anappropriate simple reparameterization, which illustrates an interesting “partial reparameterization” variantof our technique.

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653

Fig. 2 Global MCS
reparameterization in
Examples 3.2 and 4.1(λ is not the result of a one-to-one mapping from the parameter space), yet clearlyprovide trivial but precise monotone comparative statics.

3.4 Relation to Classic Supermodularity Results
We now derive a well-known supermodularity result as a particular case of Theo2
rem 3.1. Under Assumption A1, we recall that f (x, t) is supermodular in x if ∂∂xfi(x,t)∂xj
is nonnegative for all (x, t) ∈ X × T and 1 ≤ i = j ≤ n. We call the function f (x, t)
2f
supermodular in (x, t) if, in addition, ∂x∂ i ∂t
(x, t) is nonnegative for all (x, t) ∈ X ×Tk
and 1 ≤ i ≤ n, 1 ≤ k ≤ m.
Corollary 3.1 Suppose that Assumptions A1–A3 hold and that f is supermodularin (x, t). Then, x(t) is nondecreasing in t.
Proof We show that x(t) is nondecreasing in each component of t = (t1 , . . . , tm ).
Supermodularity in (x, t) implies that all components of K(x, t) are nonnegative
on X × T . It also implies that Hij (x, t) ≥ 0 for all i, j ∈ {1, . . . , m} with i = j on
X × T . Without loss of generality we can restrict our attention to the reduced feasible
set R(t) = (C ∩ D)(t). Hence Hii (x(t), t) ≤ 0 for all i ∈ {1, . . . , n}, for the Hessian
matrix is negative definite at the optimum. Since Hij−1 = (−1)i+j det(H j i )/ det(H ),it is a simple linear algebra exercise to verify that Hij−1 ≤ 0 on X × T for all i, j
in {1, . . . , n}.16 For any vector v(t) > 0, all entries of −H −1 (x, t)K(x, t)v(t)are therefore nonnegative. As a result, Assumption A5 is satisfied for ϕ(x) = x
and v(t) ≡ ek where ek is the k-th unit vector in the canonical basis of Rm . An application of Theorem 3.1 with ϕ(x) = x concludes the proof.


16 The adjoint matrix H ij is obtained by removing the i-th row and the j -th column from H .

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An important case that is not currently dealt with in the monotone comparative
statics literature is when f is supermodular in x but does not have the single-crossing
property in (x, t).17 In that context, Assumption A5 can be simplified as follows.
Assumption A5 For each t ∈ T , there exists a vector v(t) ∈ Rm such that
K(x, t)v(t) is nonnegative for all x ∈ R(t), where R(t) ⊆ X is an admissible reduced feasible set.
Corollary 3.2 Suppose that Assumptions A1–A3 and A5 hold and that f is supermodular in x. Then, x(t) is nondecreasing along the trajectories of v.
Proof We modify the proof of Corollary 3.1. Supermodularity in x ensures that
Hij−1 ≤ 0 on X × T for all i, j ∈ {1, . . . , n}. This together with Assumption A5 implies that all entries of −H −1 (x, t)K(x, t)v(t) are nonnegative. A direct applicationof Theorem 3.1 with ϕ(x) = x concludes the proof.
Corollary 3.2 applies to situations in which there are complementarities between
the different decision variables, but not between decision variables and parameters.
Since the maximizer under the original parameterization can be nonmonotonic, it isclear that Assumption A5 relaxes the tight single-crossing requirement put forwardin [6] for the price of an MCS reparameterization of the problem.
3.5 Constrained Optimization Problems
Assumption A2 requires that the optimizer be in the interior of the feasible set X .
This assumption can be relaxed in different ways, either by reducing the dimensionality of the decision space (using a substitution approach for equality constraints), orby augmenting the dimensionality of the decision space (using a Lagrange-multiplierapproach for equality and/or inequality constraints).
Substitution Approach Any equality constraints that are part of the definition of thefeasible set X in fact define a lower-dimensional set X that forms a submanifold of X
(with or without boundary). If the equality constraints can be solved globally for anumber of decision variables, the parameterized optimization problem can be viewed
as unconstrained on X after backsubstitution of these variables. More specifically,
if X is diffeomorphic to an open subset of Rn̂ with n̂ < n, the problem can be seen
as unconstrained on an open subset of Rn̂ . To render our discussion precise, consider
the problem18
max f (x, t),x∈X (t)(14)

17 In particular, f is not supermodular in (x, t).
18 Note that in this formulation it is possible to have the feasible set depend on parameters. In contrast

to standard MCS results obtained on lattices, we do not assume at the outset that X (t) is monotone in twith respect to the Veinott set order [6]. We are grateful to Pete Veinott for pointing out that his set order(originally termed “lower than” relation) was first introduced by him in a 1965 unpublished paper.

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655

with
X (t) = {x ∈ Y : g(x, t) = 0},
where Y is an open subset of Rn , t belongs to an open set T of Rm , and g takes
values in Rk (for some 1 ≤ k < n) and is twice continuously differentiable. Supposethat the level set g(x, t) = 0 can be expressed explicitly as (xn−k+1 , . . . , xn )(t) =g̃(x1 , . . . , xn−k , t), for (x1 , . . . , xn−k ) ∈ X , where X is the projection of Y on theplane {(x1 , . . . , xn−k , 0, . . . , 0) : (x1 , . . . , xn−k ) ∈ Rn−k } ⊂ Rn . The problem is thenreduced to the following (n − k)-variable unconstrained problem on X :max(x1 ,...,xn−k )∈X=

f˜(x1 , . . . , xn−k , t)max

(x1 ,...,xn−k )∈X

f (x1 , . . . , xn−k , g̃(x1 , . . . , xn−k , t), t).

The application discussed in Sect. 5.3 provides an example of this transformation.

Lagrange-Multiplier Approach Equality constraints also can be approached with
Lagrange multipliers. In the previous example, a necessary condition ([9], p. 255) for
optimality is the existence of a k-dimensional vector ν such that, at the optimum,∇x f (x, t) + ν T ∇x g(x, t) = 0.
Together with the k equations g(x, t) = 0, this determines the system of n + k equations in n + k + m variables x, t, ν,G(x, t, ν) =∇x f (x, t) + ν T ∇x g(x, t)= 0.
g(x, t)

The implicit function theorem implies that, if Gxν is invertible, then locally∇t (x, ν)(t) = − G−1xν Gt (x, ν, t).
Even though the position of the optimal x and ν in X × Rk is unknown, it might bepossible to find directions in the parameter space such that x(t) is nondecreasing inthese directions. The following example illustrates this Lagrange-multiplier approachwith equality constraints.
Example 3.3 Consider an economy with two goods (x, y) ∈ R2+ , with the production frontier {(x, y) ∈ R2+ : g(x, y) = x 2 + y 2 − 1 = 0} and a representative agent
with utility f (x, y) = u(x) + tv(y), where u, v are twice continuously differentiable,
increasing and concave. We would like to determine the monotonicity properties ofthe optimizer (x, y)(t) with respect to the parameter t. The constraint set is clearly
not a lattice, hence classic supermodularity results do not apply directly.19 Using the
19 However, classic results could be applied in conjunction with the substitution approach described earlier.

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Lagrange-multiplier approach, we have⎡⎤u (x) + 2νx⎢⎥G(x, y, ν, t) = ⎣ tv (y) + 2νy ⎦ ,x2 + y2 − 1which implies that⎡u (x) + 2ν⎢0
Gx,y,ν (x, y, ν, t) = ⎣2x
and that0
tv (y) + 2ν
2y

⎤
2x
⎥
2y ⎦
0

⎤
0
⎥
⎢
Gt (x, y, ν, t) = ⎣ v (y) ⎦ .
0⎡

An application of the implicit function theorem then yields
⎡ ⎤
⎡
⎤
−4xy
x
d ⎢ ⎥ v (y) ⎢⎥4x 2⎣y ⎦ =⎣⎦,dtDν
2y(u (x) + 2ν)
where D = −4y 2 (u (x) + 2ν) − 4x 2 (tv (y) + 2ν). In this problem, the condition ∇f + ν∇g = 0 implies that ν is negative, since the gradients of f and g bothbelong to the positive orthant of R2 . This, along with the concavity of u and v, implies that D is positive and that ∇t x < 0 < ∇t y, i.e., monotone comparative statics
obtains.
Inequality constraints can be approached in a similar fashion. Consider again problem (14), this time with
X (t) = {x ∈ Y : g(x, t) = 0, h(x, t) ≤ 0} ,where Y is an open subset of Rn , t belongs to an open set T of Rm , g and h takevalues in respectively Rk and Rr (with k < n) and are both twice continuously differentiable. The Kuhn-Tucker necessary optimality conditions ([9], p. 284) imply theexistence of adjoint variables ν and μ in respectively Rk and Rr+ , such that∇x f + ν, ∇x g + μ, ∇x h = 0andμ̂i (x, μ, t) ≡ μi (t)hi (x, t) = 0,

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657

for all i ∈ {1, . . . , r}. Letting⎡⎢
G(x, ν, μ, t) = ⎣∇ x f + ν T ∇x g + μ T ∇x hg(x, t)μ̂(x, μ, t)⎤⎥⎦,

a necessary optimality condition is that G(x, ν, μ, t) = 0. This defines a systemof n + k + r equations in n + k + r + m variables, so that (if the relevant matrix
is invertible) we can apply the implicit function theorem to compute ∇t (x, ν, μ),and proceed as in the equality case. In some problems, it is possible to know in advance which inequality constraints are binding at the optimum. In this simple case,non-binding inequalities are ignored, while binding ones are treated as equality constraints. This approach is illustrated in the applications of Sects. 5.2 and 5.3.

4 Finding a Global MCS Reparameterization
Given a smooth solution v : T → Rm to the MCS Problem, it is interesting in practice
to find an MCS reparameterization of the optimization problem (1). The idea is tostart with the flow θ : F → T induced by the vector field v and note that this flow issmooth and unique on what we refer to as the maximum “flow domain” F ⊂ R × T ,beyond which the integral curves of the vector field cannot be extended. By taking
a plane that is transverse (i.e., never collinear) to these integral curves, it is possible
to construct new parameter coordinates under which monotone comparative statics
obtain, at least locally.
4.1 Global FlowsAs a consequence of the standard theory of ordinary differential equations (ODEs)[17], if the solution v is smooth, then integral curves to the vector field exist, areunique, and induce a smooth local flow θ .20 Intuitively a flow describes how any pointof the set T is transported by a vector field v that is defined on T . To formally define
the well-known concept of a local flow, let us first introduce a flow domain F ⊂ R×Twith the property that, for any t ∈ T , the set
F (t) = {λ ∈ R : (λ, t) ∈ F} ⊂ R
is an open interval containing zero. A local flow on T is a continuous map θ : F → T
that satisfies the two group laws,θ (0, t) = t,(15)

θ (λ, θ (μ, t)) = θ (λ + μ, t),(16)

for all t ∈ T , and

20 Existence and uniqueness of integral curves is also obtained when the vector field v merely satisfies a

Lipschitz condition (cf. footnote 11).

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for all λ ∈ F (t) and μ ∈ F (θ(λ,t)) such that λ + μ ∈ F (t) . In addition, for a given local
flow θ , we define the orbits
θλ (t) = θ (t) (λ) = θ (λ, t),for (λ, t) ∈ F . A local flow (sometimes also referred to as a “one-parameter group action”) relates the vector field v(t) for any t ∈ T to its orbits θ (t) (λ). If the flow domain
is such that the map θ is surjective (i.e., θ (F) = T ), then we call θ a global flow. The
following result is standard, and will be of great use in the ensuing developments.
Theorem 4.1 (ODE Existence, Uniqueness, and Smoothness) Let v : T → Rm be asmooth vector field. Consider the initial-value problem∇λ γt (λ) = v(γt (λ)),γt (λ0 ) = t.

(17)(i) E XISTENCE : For any λ0 ∈ R there exist a nonempty open interval I which contains λ0 and an open set U ⊂ T , such that for any t ∈ U there is a smooth
integral curve γt : I → T which solves (17) for all λ ∈ I.
(ii) U NIQUENESS : Any two smooth solutions to (17) agree on their common domain.
(iii) S MOOTHNESS : Let F = I × U as in (i). If we define the local flow θ : F → T
with θ (λ, t) = γt (λ), then θ is smooth.
Proof See e.g., [18], pp. 452–459.

In the following we use global flows corresponding to the solution vector field vof the MCS problem on T (or a subset thereof) in order to find a global MCS reparameterization of problem (1) as previously indicated in Sect. 2
4.2 Coordinate Change in Parameter Space
By construction, if v is a solution to the MCS problem on T , it is nonsingular everywhere. The lack of singular points allows us to give a canonical, local representationof v using a change of coordinates in T . Consider a point t0 of T . If v is smooth,there exists an open interval I ⊂ R containing the origin and an open subset U of
T satisfying the conclusions of Theorem 4.1 (with λ0 = 0). Moreover, since v(t0 ) is
nonsingular, it uniquely determines an orthogonal hyperplane H ⊂ Rm containing t0 .
Let
P = U ∩ H ⊂ Rm .
Since H is diffeomorphic to Rm−1 , P can also be seen as an open subset of Rm−1 .
To avoid confusion, let= π(P) denote the image of P under the diffeomorphism π : H → Rm−1 . Theorem 4.1 implies the existence of a smooth flow θ (λ, t),which we restrict to the domain I × P. The flow can be reparameterized, with aslight abuse of notation, by θ(λ, π) on the product S = I × ⊂ Rm . Moreover, thecorresponding range T̄ = θ (S) is an open subset of T by Theorem 4.1. Last, θ is
one-to-one and smooth from S to T̄ , also by Theorem 4.1. We have therefore defined a local change of coordinates around t0 : any t in the open neighborhood T̄ of t0

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659

can be uniquely expressed by a tuple (λt , π(ψ(t))) ∈ S. The (m − 1)-dimensional
component ψ(t) of t is the intersection of the trajectory going through t with thehyperplane H. Moreover, since t = γψ(t) (λt ), Theorem 3.1 implies that ϕ ◦ x is nondecreasing along the trajectories of the flow θ as λ increases. We have thus proved
the following result.
Theorem 4.2 (Local MCS Reparameterization) If v solves the MCS problem on T ,there exists a local MCS reparameterization of the forms = (s1 (t), s2 (t), . . . , sm (t)) = (λt , π(ψ(t))),(18)

around any t0 ∈ T , such that ϕ(x(s)) is nondecreasing in s1 .
In practice, this result often can be applied globally, as the following example
illustrates.
Example 4.1 In Example 3.2 we obtained a vector field of the form v(t) = (−t2 , αt1 )(for some α > 0) as the solution to an MCS problem on some T ⊂ R2+ \ {0}.
Since v2 (t) = αt1 = 0 on T , we can choose the plane P = {t ∈ T : t = (t1 , 0)}, which
is transverse to the vector field. The (global) flow of the vector field v(t) is obtainedby solving the initial-value problem (17), whence
√

√
√
√
t2 sin αλ √γt (λ) = θλ (t1 , t2 ) = t1 cos αλ −, t1 α sin αλ + t2 cos αλ .
√αThus, for any (s, 0) ∈ P, we obtain√√√θλ (s, 0) = (s cos αλ, s α sin αλ),√for λ ∈ (0, π/(2 α)) and s > 0. Hence, on any contractible21 compact subset T̄
2of T , we obtain the global reparameterization t → (λ, s) with λ = √1α arctan √tαt1
22
and s = t1 + (t2 /α); cf. Fig. 2. In the context of Example 3.2, the key insight forthe decision maker from the MCS reparameterization is that the optimal number ofmarkets to invest in varies monotonically in the ratio p/q, i.e., the product price inrelation to the risk of liability lawsuits.
We now formally generalize the reparameterization technique used in the local
case by providing a general condition under which Theorem 4.2 holds globally.
Assumption A6 (Transverse Hypersurface) There exist subsets T̄ and P of T , withT̄ open and P ⊂ T̄ , such that:(i) There exists a convex, open subset
ping P onto .

of Rm−1 and a diffeomorphism π map-

21 An m-dimensional open set with nonempty interior is contractible if it is homotopy equivalent (i.e., it

can be deformed via a continuous transformation) to an m-dimensional open ball. Intuitively, contractible
sets have no “holes.”660

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Fig. 3 Global MCS reparameterization(ii) For each t ∈ T̄ , γt (F (t) ) ∩ P is a singleton {ψ(t)}.
This last assumption ensures that trajectories of an MCS vector field v lead to afoliation of the subset T̄ of T . In other words, the existence of a set P of points, eachelement of which can be associated with exactly one trajectory, allows projection of
the set T̄ onto P and—via the length (from t to ψ(t)) of the trajectory (which could
pass outside T̄ )—obtain a bijection between T̄ and a set S ⊂ Rm . This bijection corresponds to the desired global MCS reparameterization containing the length of the
MCS trajectories as one new parameter in which the solution to the reparameterized
problem (4) (when evaluated with ϕ) varies monotonically.
Theorem 4.3 (Global MCS Reparameterization) If the vector field v is a solution to
the MCS problem on T̄ and if Assumption A6 holds, then there exists a global MCS
reparameterization of the forms = (s1 (t), s2 (t), . . . , sm (t)) = (λt , π(ψ(t)))(19)

such that ϕ(x(s)) is nondecreasing in s1 .
Proof Each element t of T̄ uniquely
determines an element ψ(t) of P and a real λtsuch that γψ(t) (λt ) = t. The set S = ψ∈P F (ψ) × {π(ψ)} is an open subset of Rm .
Moreover, the mapping t → s = (λt , π(ψ(t))) ∈ S is one-to-one from T̄ to S, andsmooth by Theorem 4.1. Last, since t (s) = (λ, ψ) follows the trajectories of v as
λ increases, Theorem 3.1 implies that ϕ(x(s)) = ϕ(x(λ, π)) is nondecreasing in
s1 = λ.
When v has a potential22 u : T → R, a good candidate for P in Assumption A6 is
any iso-potential that crosses all trajectories.23 Although this need not always be thecase, it is likely that iso-potentials will be diffeomorphic to an open subset of Rm−1 ,and to a convex subset if one chooses T̄ carefully. The following classic theorem
22 That is, v is the gradient of u.
23 In that case, trajectories cross P only once, because the potential increases along them.

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661(essentially Poincaré’s lemma) gives a necessary and sufficient condition for the existence of a potential, which can be tested on any vector field v satisfying Assumption A5.
Theorem 4.4 (Existence of a Potential) Let T̄ ⊂ T be a contractible compact domain with nonempty interior and let v be a vector field on T̄ . Then, ∂vk /∂tl =∂vl /∂tk , for all k, l ∈ {1, . . . , m} with k = l, if and only if there exists a twice continuously differentiable potential u : T → R, i.e.,v(t) = ∇t u(t),for all t ∈ T .


Proof See [10], Vol. II, p. 296.

5 Applications
5.1 Optimal Capacity Choice and Product Distribution
Suppose that a firm can choose the quantity (or, equivalently, the quality) q of a product that will be provided to each of m geographically dispersed consumers. Eachconsumer k ∈ {1, . . . , m} is located at a point tk ∈ R, representing his or her type. To
deliver the product to consumer i the firm incurs a quadratic transportation (or, customization) cost d(z − tk )2 , where z ∈ R is the location of a distribution center thatthe firm is able to determine freely, and d is a positive constant. The firm’s unit transportation cost from its factory (located at the origin) to the distribution center is givenby the smooth convex function C(z) with C (0) = C(0) = 0. To keep our analysis
simple, we assume that each consumer’s demand can be represented by a linear inverse demand function of the form a − bq, where a, b are positive constants.24 The
firm’s profit can therefore be written as(q, z, t) = mq(a − bq) − dqm(tk − z)2 − mqC(z),

(20)

k=1

where t = (t1 , . . . , tm ) ∈ Rm . The firm thus solves the parameterized optimization
problemmax(q,z)∈R+ ×R(q, z, t).

(21)

Provided that a is sufficiently large, it is easy to verify that problem (21) has a uniqueinterior solution (q ∗ (t), z∗ (t)) and satisfies Assumptions A1–A3. The firm’s MCS
problem is: how does the optimal per-consumer production quantity q ∗ vary with t?
24 This corresponds to a quadratic utility function u (q) = aq − bq 2 /2 for all consumers k ∈ {1, . . . , m}.
kAllowing for demand heterogeneity with uk (q) = ak q − bk q 2 /2 leads to analogous results.

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Starting from a market characterized by the parameter vector t ∈ Rm , is there a direction in the parameter space in which the optimal production quantity q ∗ increases?
We first determine the first-order necessary optimality conditions for (21),∂= a − 2bq − d(tk − z)2 − mC(z) = 0
∂q

(22)


∂
= 2dq
(tk − z) − mqC (z) = 0.
∂z

(23)

m

k=1

and
m

k=1

Therefore, we can restrict R(t) to the subset of tuples (q, z) ∈ X = R+ × R thatsatisfy (22) and (23). To apply the method, we also compute , H −1 and K. Sinceϕ(q, z) = q, we have that = (1, 0) and Assumption A4 is satisfied. Moreover,
H=

2d

m

−2b

2d

k=1 (tk − z) − mC (z)m

k=1 (tk − z) − mC (z)−2mdq − mqC (z).

The first-order necessary optimality condition with respect to z, given in (23), simplifies the Hessian matrix H to
H=

−2b

0

0

−2mdq − mqC (z).

It follows that25
H −1 =

1
2bmq(2d + C (z))−2mdq − mqC (z)0whence− H −1 =The first row of K is given by26
 2
∂
∂2
,...,
∂q∂t1
∂q∂tm



0
,
−2b

1
,0 .
2b

= 2d ((z − t1 ), . . . , (z − tm )) .

Therefore, in order to satisfy Assumption A5, we are looking for a vector v(t) ∈ Rm
such thatd(z − t1 , . . . , z − tm ), v(t) ≥ 0,b
25 Observe that the determinant is nonzero, since C (z) ≥ 0 by convexity of g.
26 There is no need to compute the second row, since K is left-multiplied by − H −1 , whose second

component is zero.

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for all z ∈ R, or equivalently
(ze − t), v(t) ≥ 0,(24)

for all z ∈ R, where e = (1, . . . , 1)/m is the unit vector of the first bisectrix27
 in Rm . It is easy to see that if v(t) is orthogonal to e and has a nonnegative
scalar product with −t, the condition is satisfied. The vector v(t) = −t + mt, ee
is such that first, e, v(t) = 0, since e, e = 1/m; and second, −t, v(t) = t, t −m(t, e)2 ≥ 0 by the Cauchy-Schwarz inequality. Moreover, the inequality is strict if
t is not collinear with e. Last, observe that, when seen from t, v(t) points directly28
to the first bisectrix  of Rm . We therefore conclude from Theorem 3.1 that q ∗ (t) increases as t gets closer to . In other words, as the consumer types become “closer,”the optimal product quantity increases. When the consumer types are identical (tcollinear to e), the optimal production reaches its maximum. The problem can thus
be reparameterized in the following way: define a cylinder P around the first bisectrix , for example
P = t ∈ Rm−1 : d(t, ) = 1 ,where d is the Euclidian distance from a point to a line. This cylinder is an (m − 1)dimensional manifold, which can be parameterized by m − 1 components. Moreover, P is transverse to all trajectories, and is hit by all trajectories once, so that
Assumption A6 is satisfied.29 Therefore, we have a global reparameterization of Rm
where m − 1 components correspond to the position on the cylinder and determinea radius emanating from , and the remaining component is a parametric representation of the radius. In this particular context, it is possible to construct a more
efficient parameterization: let H denote the (m − 1)-dimensional hyperplane of Rm
orthogonal to the first bisectrix and going through the origin, and (1 , 2 , . . . , m−1 )be an orthogonal basis of H. Then, (e1 , e2 , . . . , em ) = (1 , 2 , . . . , m−1 , e) is an orthogonal basis of Rm . Moreover, if t is represented with respect to that basis, i.e.,t = s1 e1 + s2 e2 + · · · + sm em , we obtain a new parameterization of the parameter
space such that q ∗ (s1 , . . . , sm ) is nonincreasing in (s1 , . . . , sm−1 ) (the smaller thesecoordinates, the closer is t to the first bisectrix).30
It is worth observing that in order to solve the MCS problem, we relied on our
knowledge of the first-order optimality condition for z. It is our second use of the
crucial device R(t) which enables us to narrow down the domain on which Assumption A5 must be satisfied. On the other hand, we did not use the fact that the optimizer (q ∗ , z∗ ) satisfies the first-order optimality condition for q. Thus R(t) could
have been larger without affecting our ability to construct the vector field v to satisfy Assumption A5. We also note that classic supermodularity is of no use in this27 The first bisectrix is defined by the equation t = t = · · · = t .
m
1
2
28 The vector v(t) points in the direction of the orthogonal projection of t on .
29 To be rigorous, P is diffeomorphic to S m−2 × R, where S m−2 is the unit sphere in Rm−1 . This para-

meterization is a generalization of cylindric coordinates in R3 .
30 Moreover, it can be shown that q ∗ (s) is independent of the last component, s .
m

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problem, since q ∗ is not monotonic in any of the tk ’s. Last, observe a remarkable factin our analysis of this example: we are able to obtain monotone comparative statics
for q ∗ without solving explicitly for either q ∗ or z∗ . In general, the method can beused to derive monotone comparative statics for any single decision variable, say, x1 ,while one is able to solve the optimization problem explicitly for some other variables, say, xk (t), xi+1 (t), . . . , xn (t). In that case the reduced feasible set R(t) can be
narrowed down to the set of all x ∈ X such that xk = xk (t), . . . , xn = xn (t).
5.2 Neoclassical Production
Consider a firm’s optimal choice of factor inputs, capital k and labor l, so as to maximize the objective function
f (x, t) = g(k, l) − rk − wl,(25)

where x = (k, l) and t = (r, w) with r the rate of return of capital and w the average
wage rate. As pointed out by [6], if g is not supermodular, comparative statics are notmonotone in the original parameterization. To demonstrate the use of our method,
we assume that g is twice continuously differentiable and that there exists a uniqueoptimizer in the interior of R2+ . Therefore, Assumptions A1–A3 are satisfied, with
the Hessian and cross-derivative matrices
H=

gkk
gkl

gkl
gll

and K =

−1 0
,
0 −1

respectively. The pseudogradient is thereforeW=1D

gll −gkl,−gkl gkk

2 )(k, l) is the determinant of H (k, l). Because of thewhere D(k, l) = (gkk gll − gkl
strict concavity of g at the optimizer, we can restrict the reduced feasible set R(t)to the subset of R2+ where H is negative definite, implying that D(k, l) is positive.31
Notice that the pseudo-gradient and the reduced feasible set are independent of r
and w.32 To simplify our exposition we drop the explicit dependence on t and referto R(t) as R. If one can find a vector v making a positive scalar product with W (k, l)
for all (k, l), it will satisfy Assumption A5 for all values of r and w. The vector field
will then consist of straight, parallel trajectories of direction v. This will generate
a linear reparameterization of the problem (the basis of the new coordinate system
consisting of v and any other vector not collinear to v), under which both k and l
are nondecreasing in the first parameter coordinate. Before addressing the problem
of monotone comparative statics for both k and l, let us consider the simpler problem

31 The determinant D is positive at any maximizer of (25) as the product of the two negative eigenvaluesof H .
32 Note that H , K and D are all independent of (r, w).

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of finding monotone comparative statics for k alone. That is, we consider the function ϕ(k, l) = k, which trivially satisfies Assumption A4. The pseudogradient thenbecomesWk (k, l) =with(k, l)W (k, l),(k, l) = (1, 0). This yields
Wk (k, l) =1
(gll , −gkl ).
D

We are looking for a vector v ∈ R2 \ {0} such that Wk (k, l), v ≥ 0, or equivalently
(gll , −gkl ), v ≥ 0.

(26)

Since gll is nonpositive, a solution is v = (−1, 0). That is, k(r, w) is nondecreasing
in r. Notice that this result obtains without assumption on g except for smoothness.33
In general, k(r, w) is not monotonic in w: this would require gkl ≥ 0 for all (k, l)(as can be seen by substituting v = (0, −1) in (26)). However, there may be otherdirections of v such that k is nondecreasing. Equation (26) can be rewritten as
v1 gll (k, l) − v2 gkl (k, l) ≥ 0,
for all (k, l) ∈ R2+ . Since gll ≤ 0 on R, this is equivalent tov1 ≤ min{D, d}v2 ,(27)

where (d, D) = (inf δ, sup ), with


gkl
δ=
(k, l) : gkl ≥ 0, gll < 0, (k, l) ∈ R ,
gll


gkl
(k, l) : gkl ≤ 0, gll < 0, (k, l) ∈ R ,=gll
as well as the conventions that inf{∅} = +∞ and sup{∅} = −∞. When  = ∅, D ≥ 0.
Similarly, d ≤ 0 if δ = ∅. When g is supermodular,  is empty or reduced to the singleton {0}, so that D ≤ 0. Moreover, δ = ∅ implies d ≤ 0, so that condition (27) is satisfied by any v ∈ R2− , by virtue of the nonpositivity of d ∧ D. This proves that k(r, w)
is nonincreasing not only in r, but also in w, whenever g is supermodular. In general,
relation (27) defines a convex cone k ∈ R2 based at the origin which always contains the negative real line R− × {0}. Except when both D = +∞ and d = −∞, k
has a nonempty interior. If gkl > 0, D = −∞, implying that k is a half-space that is
located below the line v1 = dv2 .
33 Another way to see this is the following: the function f (k, l, r, w) = g(k, l) − rk − wl is supermodularin (r, k). To apply standard supermodularity results, define F (k, r, w) = maxl≥0 f (k, l, r, w). F is supermodular in (k, r) and arg maxk≥0 F (k, r, w) = k(r, w). This implies that k(r, w) is nondecreasing in r. We
thank Paul Milgrom for this observation.

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The optimizer k(r, w) is nondecreasing in any direction of k . When k has a nonempty interior, it is possible to change coordinates in the parameter space by using
two independent basis vectors in k . As pointed out earlier, this coordinate change isglobal, since k is independent of the particular values chosen for r and w. Similarly,l(r, w) is nondecreasing in any direction located in the cone l based at the origin
and containing the negative imaginary line {0} × R− . Having constructed these twocones, we can now address the more challenging question, is it possible to find directions in which both k and l increase? The answer depends on V = k ∩ l . If V isempty, we cannot construct any direction that jointly increases k and l. If V is nonempty, then it is also a convex cone, whose elements are directions of joint increase.
The intersection V being empty does not prove the nonexistence of directions of joint
increase. It just means that we do not have enough information on the optimizers
to produce such directions. As our information gets richer, the set R becomes narrower, which implies that the cones k and l become wider. When one has enoughinformation, the cones are wide enough to intersect, yielding the desired directionsof joint increase (cf. Fig. 4).34 When V is nonempty, the vector field can be chosen
constant: v(r, w) = v for some v ∈ V. In that case, any straight line P orthogonal to v
satisfies Assumption A6: it is transverse and is hit exactly once by all trajectories.
The reparameterization is then simple: take any vector e2 on that line and let e1 = v.
Then, (e1 , e2 ) is an orthogonal basis of R2 , such that if one expresses t = (r, w) onthat basis (that is, (r, w) = s1 e1 + s2 e2 ), then (k(s), l(s)) is nondecreasing in s1 .
Fig. 4 Neoclassical production:
monotonicity of k(r, w) onk ∩ l

34 The analysis for obtaining directions of joint decrease is naturally analogous, by considering opposite

directions. Similarly, it is possible to obtain directions of increase in one parameter and decrease in theother.

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5.3 Giffen Goods
In an economy with two goods, an agent wishes to maximize her increasing andconcave utility by solving
max u(x, y),(x,y)∈R2+

subject topx + qy ≤ w,
where x, y are the quantities of the two goods, p and q are their respective positive
prices, and w represents the agent’s wealth. Without loss of generality, we select thesecond good to be the numéraire and correspondingly set q = 1. In addition, sinceboth goods are desirable, the agent’s budget constraint is binding. As pointed out in
Sect. 3.5, the agent’s problem can be restated asmax

x∈[0,w/p]u(x, w − px).

In this formulation the problem has one decision variable and two parameters.35 If u
is smooth, compactness ensures the existence of an optimizer. We assume that forour starting values of w and p, the optimizer x(w, p) is unique, located in (0, w
p ),and that u(·, w, p) is locally strictly concave at x(w, p) and everywhere twice continuously differentiable. This implies that Assumptions A1–A3 are satisfied. The firstgood is “normal” if x(w, p) is nonincreasing in p, and Giffen or “inferior” if thismonotonicity is sometimes violated.36 Similarly, one would usually expect the consumption of any good to increase with the agent’s wealth. However, this monotonicity is also sometimes violated. The question then becomes, under what conditions isa good normal, and how are price and wealth effects connected? The Hessian andcross-derivative matrices are
H = uxx − 2puxy + p 2 uyy

and K = uxy − puyy − xuxy + pxuyy − uy .

The pseudogradient W is therefore given by
W (x, w, p) =1[α(x, w, p) − xα − uy (x, w − xp)],
D

where D = −H > 0 (by strict concavity of H at the optimizer) and α(x, w, p) =uxy − puy y. We also note that since ϕ(x) = x, Assumption A4 is trivially satisfied.
35 As mentioned earlier, we have therefore converted the initial problem—with two decision variables on

a one-dimensional manifold of R2 (with empty interior but nonempty relative interior)—into a problem
with one decision variable on a set with nonempty interior.
36 Classic examples include potatoes or bread. The gist of the argument goes as follows: when the price forbread increases, poorer people cannot afford buying “luxury goods” such as meat, and end up consuming
more bread, which is still the cheapest good. Other goods violating this monotonicity are Veblen goods [19]or positional goods ([20], Chap. 3) which are such that the implied status of the owner increases with theirprice.

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In order to meet Assumption A5, we are thus looking for a vector v ∈ R2 such thatα(x, w, p)v1 − (xα(x, w, p) + uy )v2 ≥ 0,(28)

for all x in the reduced feasible set R(w, p). First, we observe that if u is supermodular and concave in its second variable, α is nonnegative, which implies, along with
the nonnegativity of uy , that any vector v in R+ × R− solves (28). This means that
if the two goods are complements and if the utility function is concave in the second
good, then the first good is normal.37 In the general case, we show that there is a hierarchical relationship between wealth and price effects. Increasing wealth amounts
to setting v1 > 0 and v2 = 0, so that the good is normal with respect to the wealtheffect if and only if α ≥ 0. On the other hand, α ≥ 0 implies that xα + uy ≥ 0.
Since decreasing the price amounts to setting v2 > 0 and v1 = 0, the good is therefore normal with respect to the price effect if α ≥ 0. This shows that the following result holds for any smooth, nondecreasing utility function: if an augmentation
in wealth increases the optimal consumption of a good, then a cut in its price also
increases its optimal consumption. In general, the converse is not true.38 If the optimal consumption x(w, p) is known or constrained to belong to some subinterval
J = (x1 , x2 ) ⊂ (0, w/p),39 the analysis can be refined. For example, suppose thatminx∈J {α(x, w, p)} ≥ 0 for all w, p. Then the good is normal with respect to both
wealth and price effects (any v ∈ R+ × R− solves (28)). If α sometimes takes negative values but minx∈J {xα(x, w, p) + uy (x, w − px)} ≥ 0, then the good is normal
with respect to price effects (v ∈ {0} × R− solves (28)). More generally, supposethat α(·, w, p) changes sign only once on (x1 , x2 ), and that α(x1 , w, p) > 0 for all w
and p in an open neighborhood of initial values of wealth and price. The second
condition means that the good is normal for low consumption, while the first condition means that the good becomes Giffen for high consumption values. Then, if thevectorsα(x1 , w, p), −x1 α(x1 , w, p) + uy (x1 , w − px1 )andα(x2 , w, p), −x2 α(x2 , w, p) + uy (x2 , w − px2 )are in the same half-plane, there exists a normal vector v(w, p) of the half-space
whose scalar product with W (x, w, p) is nonnegative for all x ∈ (x1 , x2 ). The situation is represented in Fig. 5.
It is easy to verify that v(w, p) can always be taken in the negative orthant R2− .
Therefore we have the following result: if the pseudo-gradients of consumption
boundaries x1 , x2 lie in the same half-plane, and if the good behaves as a normal
good for low consumption values and as a Giffen one for high consumption values
37 This result can also be shown by observing that the concavity of u in y implies the supermodularity of v

in (x, w, −p).
38 The result can be read in the opposite direction: if a good is inferior with respect to price effect, it is alsoinferior for wealth effect.
39 For example, minimal consumption could be imposed or supply could be limited.

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Fig. 5 Giffen goods: solving
the MCS problem for x(w, p)(such as potatoes, cf. footnote 36), then there exists a way to increase optimal consumption of this good by reducing both wealth and price at the same time. It canalso be shown that in the same situation, there is no way to increase consumption byraising both wealth and price at the same time.
5.4 Multiattribute Screening
A variation of our method can be used in the context of screening with multiple instruments. A firm faces customers of different types, distributed on an interval X ⊂ R
according to a positive density function g. The firm sells products whose m attributes
are described by a vector t ∈ T ⊂ Rm . The goal of the firm is to propose a product line  ⊂ T and a price schedule P :  → R+ (with 0 ∈  and P (0) = 0) thatmaximizes its expected profitπ(P , ) =[P (t (x)) − C(t (x))] g(x)dx,
X

where C(t (x)) is the cost of producing t (x) ∈  and t (x) solves the type-x consumer’s utility maximization problem
t (x) ∈ arg max {u(x, t) − P (t)} .
t∈This general screening problem has been only solved for particular cases.40 Whenm = 1 (only one instrument available), it is possible to directly compute the optimal price schedule P (x) under some additional supermodularity assumptions on the
40 See [21–23] for the multiattribute, one-dimensional type case, and [24] for a recent account of thegeneral multidimensional screening literature.

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primitives of the problem. Assuming that u is smooth, we can define
μ(x, t) = u(x, t) − ux (x, t)

1 − G(x),g(x)

where G is the probability distribution of the density g. [25] has shown, based on atechnique developed by [26], that if u and μ are twice differentiable and supermodular (i.e., uxt ≥ 0 and μxt ≥ 0 on X × T ), then it is possible to construct the optimal
price schedule. Moreover, this optimal schedule leads to “perfect” screening (withoutbunching): each consumer of type x will buy a distinct product t (x). Suppose now
that there are m ≥ 2 product attributes available. Our method can be extended to build
product lines that perfectly screen consumers. Defining the pseudogradient W as the(2 × m)-matrix
W (x, t) =∇xT ut (x, t)∇xT μt (x, t),

suppose that there exists for all t ∈ T a nonzero vector v(t) such that
W (x, t)v(t) ≥ 0,
for all x ∈ X (i.e., Assumption A5 is satisfied). We can then define  ⊂ T to be
the image of any smooth trajectory γ : (0, 1) → T generated by the vector field v.
This leads to a reparameterization of the utility u and the function μ when restricted
to X ×  (that is, when customers are offered the product line ). Specifically, wedefine ũ and μ̃ on X × (0, 1) by ũ(x, λ) = u(x, γ (λ)) and μ̃(x, λ) = μ(x, γ (λ)).
Using Lemma 3.1 and Theorem 3.1, we can show that ũ and μ̃ are supermodular
on X × (0, 1). The aforementioned result then implies that it is possible to find the
optimal price schedule on , and that this schedule perfectly screens customers. Thisapproach does not solve the original problem of maximizing the profit on T , since
we artificially restricted ourselves to the product line . However, the method canbe repeated for several distinct trajectories, and leads to a perfectly screening priceschedule that maximizes the expected profit not only on a particular product line, but
on a large class of product lines that spans the whole multiattribute space T .

6 Discussion
In the available literature on monotone comparative statics, the parameterization of
the optimization problem is essentially taken as given.41 The presently known criteria
for the monotonicity of solutions hold, therefore, only with respect to the particular
problem formulation given at the outset. Milgrom and Shannon’s [6] characterizationof the monotonicity of solutions to (1) on lattices requires the objective function f to
41 Note that the decision variables are also typically taken as given. Our method in principle allows for achange of the decision variables to obtain monotone comparative statics through an appropriate choice ofthe evaluation function ϕ.

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be quasi-supermodular in x and to satisfy a single-crossing property in (x, t).42 The
supermodularity requirement on the objective function can thereby be interpreted interms of “complementarity” of decision variables, a concept that dates back at least to
Edgeworth [28] and whose origins are reviewed by [29]. Reference [30] demonstrates
the power of complementarities and associated supermodularity properties in interpreting decision changes as monotone responses to exogenous shifts of economic
conditions. Even though equilibria cannot be located exactly, complementarities allow one to make precise statements about the direction in which optimal decisions
move as a consequence of parameter changes. In the absence of such complementarities, the presently available theory unfortunately guarantees the non-monotonicity ofsolutions, even though this non-monotonicity might just be a symptom of an unsuitable parameterization of the problem. This paper proposes a way to obtain an equivalent formulation of the optimization problem (1) using a new parameterization, suchthat—provided sufficient knowledge about the location of the solution—monotone
comparative statics may be obtained. Finding a new parameterization of the problem
amounts to creating a set of economic indicators which allow for monotonic decision making and thus easy rules of thumb (i.e., when the relevant indicator goes up,
the optimal decision goes up, too). This seems especially useful in situations where
the same optimization problem needs to be solved repeatedly for different parametervalues.
Let us briefly mention at this point that our method naturally extends to equilibrium problems (cf. also [31]) specified by a relation
F (x, t) = 0,
where F : X × T → Rn is a continuously differentiable function, as can be seen by
associating ∇x f with F , so that H = ∇x F and K = ∇t F . With these substitutions
in place, all of our results hold essentially without modification. We also remind
the reader that, as pointed out in Sect. 3.5, even though through Assumption A2
we require the existence of a unique interior global optimum, parameter-dependent
constraints can be accommodated in a straightforward way by shifting the analysis
to a submanifold in X or by augmenting the space of decision variables by Lagrange
multipliers corresponding to the binding constraints.
Sometimes our method may also be useful for reducing the number of parameters
without any losses. To show this, let us first note that, clearly, an “ideal” parameterization of problem (1) for the case m ≥ n associates exactly one parameter ti witheach component of the decision variable xi , and is such that xi remains unchanged
in response to a change of parameter tj (with j = i). If the location of the optimal
solution is perfectly known, a reparameterization with these “ideal” properties canalways be obtained by setting si = xi (ti ) for i ∈ {1, . . . , n} and simply discarding allother m − n parameters, tn+1 , . . . , tm . Unfortunately, lack of knowledge about thelocation of the optimizer (up to a monotone transformation) usually makes this trivial solution impracticable. Nevertheless, it may sometimes be possible to reduce the42 Reference [27] applies these results to expected-utility maximization problems under uncertainty andfinds necessary and sufficient conditions on the model primitives in that context.

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number of parameters by finding directions v(t) for which W (x, t), v(t) = 0 for
all x in an admissible reduced feasible set R(t). If such a direction can be found, thesolution does not depend on λ in the associated global MCS reparameterization and λmay thus be discarded from the set of new parameters. By repeating this process it
may be possible to eliminate further parameters.
The tradeoff between the decision maker’s knowledge about the location of the
optimal action and her ability to find directions that guarantee monotone behavior of
the optimizer (i.e., to solve the (local) MCS problem) is related to “partially specified
problems”, as discussed by [13]. One type of partially specified optimization problems possesses an objective function of the form f (x, t) = g(x, t) + δ(x), where δis any affine mapping from X ⊂ R to R. Monotonicity of optimal solutions xδ (t)to the problem (1) for any affine δ is then equivalent to the (otherwise unknown)function g being supermodular on X × T . The key idea in this approach is that theclass of perturbations δ is large enough relative to g and X to allow for any location of the optimizer in X . A variation of our method bypasses this definitive result
when the function g fails to be supermodular: it might be possible to transform theparameter space so as to “supermodularize” the function g. If g(x, t) is not supermodular in (x, t), we can build trajectories γ : (0, 1) → T in the parameter spacesuch that g(x, γ (λ)) is supermodular in (x, λ). We have also used this approach in
Sect. 5.4 and Corollary 3.2. While supermodularization of functions is just a particular application of our method, its repeated use in this paper suggests its potential
benefits in numerous other settings, such as for supermodularizing noncooperative
games.
Further research could proceed to relax some of the differentiable structure imposed to obtain our results. Systematic MCS reparameterizations can be expected
to generalize naturally to an analysis on lattices. The problem is to find a ruleon W ⊂ T × T such that (t, t ) ∈ W implies φ(x(t )) ≥ φ(x(t)). In particular, suppose that we can build trajectories {Ti } in T such that (x, t) → f (x, t) has the singlecrossing property43 on X × Ti for all i ∈ I. If f is in addition (quasi-)supermodular
in x, standard results as in [13] apply, showing that x(t) is nondecreasing along the
trajectories. The problem is of course to construct such trajectories. Our method for
doing so is based on differential calculus, but there may be other ways to build trajectories (e.g., through discretization of our results), or at least to find rules in theparameter space, in order to achieve monotone comparative statics.

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